FOURTH EDITION

Physical Pharmacy-

PHYSrCAL CHEMICAL PRINCIPLES I~ THE PHARMACEUTICAL SCIENCES

Alfred Martin, Ph.D. Emeritus CoulterR. Sublett Professor

Drug Dynamics Institute, College of PluLrmacy, University of Texas

with the participation of PILAR BUSTAMANTE, Ph.D.

Titular Professor Department.of Pharmacy

and Pharmaceutical Technology, University Alcala de Henares,

Madrid, Spain

and with illustrations by A. H. C. CHUN, Ph.D. Associate ResearchFellow

PharmaceuticalProducts Division, Abbott Laboratories

B. I. Waverly. Pvt Ltd New Delhi

B. I. WaverlyPvtLtd 54 Jan.-til, New Del.. - 110 001

Reprintauthorisedby Waverly International

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Indian Reprint 1994 Reprintl99S

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Price h. 495.00

Publishedin India by B.I. Waverly Pvt Ltd, 54 Janpatb, New Delhi - 110001 aDd prill1ed at United IDdia Press, New Delhi.

Dedicated to my parents Rachel and Alfred Martin, Sr., my wife, Mary, and my sons, Nen' and, Douglas.

Preface

The fourth edition of Physical Pharmacy is concerned, as were earlier editions, with the use of physical chemicalprinciples as applied to the various branches of pharmacy. Its purpose is to help students, teachers, researchers, and manufacturing pharmacists use the elements of mathematics, chemistry, and physics in their work and study. The new edition has been updated and revised to reflect a decade of current advances, concepts, methods, instrumentation and new dosage forms and delivery systems. Two chapters in the third edition-Introductory

Calculus and Atomic and Molecular Structure-have been removed. The calculus chapter has been replaced by an appendix that provides necessary rules ofdifferentiation and integration. The space made available by these deletions has allowed extensive revision in other chapters: Complexation and Protein Binding, Kinetics, Interfacial Phenomena, Colloids, Rheology, and Coarse Dispersions. The chapter on Drug Product Design has been rewritten and expanded to reflect the many advances in controlled drug delivery systems over the past two decades. The problems at the end of the chapters have been varied and considerably increased in number. * This new and revised edition will bring readers up-to-date with the last lQ years of progress in the physical and chemical foundations of the pharmaceutical sciences. The author acknowledges the outstanding contribu

tions of Professor Pilar Bustamante to the preparation of this fourth edition with regard to originating new problems and writing a major part of Chapter 19, Drug Product Design. The time and professional devotion she gave to the revision process, in a variety of ways, was exceptional. Dr. A. H. C. Chun prepared most of the

illustrations, as he has skillfully done for each of the editions. Dr. Stephen-Baron, who checked the problems in the first edition, has again assisted in reviewing the problems and in reading galley proof for the fourth edition. The author expresses pis appreciation for additional

contributions to this book by Dr. R. Bodmeier, University of Texas; Dr. Peter R. Byron, Virginia Commonwealth University; Dr. S. Cohen, Tel-Aviv University, Israel; Dr. T. D. J. D'Silva, Rhone-Poulenc; Dr. J. B. Dressman, University of Michigan; ,Dr. J. Keith Guillory, University of Iowa; Dr. V. D. Gupta, University of Houston; Dr. Bhupendra Hajratwala, Wayne State University; Dr. E. Hamlow, Bristol-Meyers Squibb; Dr. A. J. Hickey, University of Illinois at Chicago; S. Jarmell, Fisher Scientific; Dr. A. E. Klein, Oneida Research Services; Dr. A. P. Kurtz, Rhone-Poulenc; pre Z. Liron, Tel-Aviv University, Israel; Dr. T. Ludden, U.S. Federal Food and Drug Administration; Dr. James McGinity, University of Texas; B. MillanHernandez, Sterling International, Caracas, Venezuela; Dr.Paul J. Niebergall, Medical University of South Carolina; Dr. Robert Pearlman, University of Texas; Dr. R. J;. Prankerd, University of Florida; H. L. Rao, Manipal, India; Dr. E. G. Rippie, University of Minnesota; T. Rossi, Fisher Scientific; Dr. Hans Schott, Temple University; Dr. V. J. Stella, University of Kansas; Dr. Felix Theeuwes, Alza Corporation; Dr. K. Tojo, Kyushu Institute of Technology, Japan; and Dr. J. Zheng, Shanghai Medical University. Recognition is also given for the use of data and

reference material found in the Merck Index, 11th Edition, Merck, 1989; the. U.S. Pharmacopeia, XXII-NF XVII, U.S. Pharmacopeial Convention, 1990;

fThe percent increase in figures, tables, and so on in the 4th edition of Physical Pharmacy as compared with those in the 3rd edition is as follows:

Figures Tables References Equations Examples Problems

0/0 Increase 12 2 45 17 107

viii Pretace

and the CRC Handbook of Chemistry and Physics, 63rdo Edition, CRC Press, 1982. The author acknowledges with thanks the use of problems patterned after some of those in J. William Moncrief and William H. Jones, Elements of Physical Chemistru, Addison-Wesley, 1977; Raymond Chang, Physical Chemistry with Applications to Biological Systems, 2nd Edition, Macmillan, 1981; and David Eisenberg and Donald Crothers, Physical Chemistry with Application to the Life Sciences, Benjamin/Cummings, 1979. The suggestions and assistance of Dean J. T. Doluisio

.andcolleagues at the University of Texas are acknowledged with thanks. Much of the work by the author was

Austin, Texas

made possible by the professorship provided by Coulter R: Sublett. Dr. Pilar Bustamante is grateful to Dean Vicente

Vilas, to Professor Eugenio Selles, and to her colleagues at the University Alcala de Henares, Madrid, Spain, for their support and encouragement during the many hours of revision work. George H. Mundorff, Executive Editor; Thomas J.

Colaiezzi, Production Manager; and Dorothy A. Di J

Rienzi, Project Editor of Lea & Febiger have been patient, sympathetic, and helpful during the revision for the fourth edition of Physical Pharmacy.

ALFRED MARTIN

Contents

1. Introd uction 1 2. States of Matter 22 3. Thermodynamics 53

4. Physical Properties of Drug Molecules 77 5. Solutions of Nonelectrolytes 101

6. Solutions of Electrolytes 125 7. Ionic Equilibria 143

8. Buffered and Isotonic Solutions 169 9. Electromotive Force and Oxidation- Reduction 190

10. Solubility and Distribution Phenomena 212 11. Complexation and Protein Binding 251

12. Kinetics 284 13. Diffusion and Dissolution 324

14. Interfacial Phenomena 362 15. Colloids 393

16. Micromeritics 423 17. Rheology 453

18. Coarse Dispersions 477 19. Drug Product Design 512

20. Polymer Science 556 Appendix: Calculus Review 595

Index 603

1 Introduction

Dimensions and Units Statistical Methods and the Analysis of Errors Some Elements of Mathematics

-----------------------_:p-----------------

The pharmacist today more than ever before is called upon to demonstrate a sound knowledgeof pharmacology, organic chemistry, and biochemistry and an intelligent understanding of the physical and chemical properties of the new medicinal products that he or she prepares and dispenses. Whether engaged in research, teaching, manufactur

ing, community pharmacy, or any of the allied branches of the profession, the pharmacist must recognize the need to borrow heavily from the basic sciences. This stems from the fact that pharmacy is an applied science, composed of principles and methods that have been culled from other disciplines. The pharmacist engaged in advanced studies must work at the boundary between the various sciences and must keep abreast of advances in the physical, chemical, and biologic fields to understand and contribute to the rapid developments in his own profession. Pharmacy, like many other applied sciences, has

passed through a descriptive and an empiric era and is now entering the quantitative- and theoretic stage. The scientific principles of pharmacy are not as

complexas some would believe, and certainly they are not beyond the understanding of the well-educated pharmacist of today. In the followingpages, the reader willbe directed through fundamental theory and experimental findings to practical conclusions in a manner that should be followed easily by the average pharmacy student. The name physical pharmacy has been associated

with the area of pharmacy that deals with the quantitative and theoretic principles of science as they apply to the practice of pharmacy. Physical pharniacy attempts to integrate the factual knowledge of pharmacy through the development of broad principles of its own, and it aids the pharmacist, the pharmacologist, and the pharmaceutical chemist in their attempt to predict the solubility, stability, compatibility, and biologic action of drug products. As a result of this knowledge, the

pharmaceutical scientist is in a better position to develop new drugs and dosage forms and to improve upon' the various modes of administration. This course should mark the turning point in the

study pattern of the student, for in the latter part of the pharmacy curriculum, emphasis is placed upon the application of scientific principles to practical professional problems. Although facts must be the foundation upon which any body of knowledge is built, the rote memorization of disjointed "particles" of knowledge does not 'lead to logical and systematic thought. The student should strive in this course to integrate facts and ideas into a meaningful whole. In the pharmacist's career, he or she frequently will call upon these generalizations to solve practical pharmaceutical problems. The comprehension of course material is primarily

the responsibility of the student. The teacher can guide and direct, explain and clarify, but facility in solving problems in the classroom and the laboratory depends largely on the student's understanding of theory, recall of facts, ability to integrate knowledge, and willingness to devote sufficient time and effort to the task. Each assignment should be read and outlined, and assigned problems should be solved outside the classroom. The teacher's comments then will serve to clarify questionable points and aid the student to improve his or her judgment and reasoning abilities.

DIMENSIONS AND UNITS The properties of matter are usually expressed by

the use of three fundamental dimensions: length, mass, and time. Each of these properties is assigned a definite unit and a reference standard. In the metric system, the units are the centimeter (em), the gram (g), and the second (sec); accordingly, it is often 'called the egs system. A reference standard is a fundamental unit

1

I1

1

2 Physical Pharmacy

TABLE 1-1. Fundamental Dimensions and Units

Dimension (Measurable Dimensional Reference Quantity) Symbol CGS Unit 51 Unit Standard Length (I) L Centimeter (em) Meter(m) Meter Mass (m) Time (t)

M --T

Gram (g) Second (sec)

Kilogram (kg) Second (s)

Kilogram Atomic frequency of Cesium 133

relating each measurable quantity to some natural or artificial constant in the universe.

Measurable quantities or dimensions such as area, density, pressure, and energy are compounded from the three fundamental dimensions just referred to. In carrying out the operation of measurement, we assign to each property a dimension that is expressed quantitatively in units. Thus the quantities of length, area, and volume are measured in the dimension of length (L), length squared (2), and length cubed (3), respectively corresponding to the unit of em, cm'', and em" in the cgs system. The fundamental dimensions, units, and reference standards are given in Table 1-1.

The International Union of Pure and Applied Chemistry (IUPAC) has introduced a Systeme International or 81 units in an attempt to establish an internationally uniform set of units. Physical Pharmacy generally uses the cgs or common system of units. However, since SI units appear with increasing frequency in research articles and are found in some textbooks, they are introduced to the student in this chapter. They are also used in Chapter 4 and to some extent elsewhere in the book. SI units are listed in Tables 1-1 and 1-2, and some appear inside the front cover of the book under Physical Constants. Length and Area. The dimension of length serves as a

measure of distance and has as its reference standard the meter. It is defined as follows:

1 meter = 1.65076373 x 106AKr_86 in which AKr-86 = 6.057~021 X 10-7 m is the wavelength in vacuo of the transition between two specific energy levels of the krypton-86 atom, Prior to this definition, the meter was arbitrarily' defined as the distance between two lines on a platinum-iridium bar preserved at the International Bureau of Weights and Measures in Sevres, France. The unit of length, the centimeter, is

TABLE 1-2. Fractions and MUltiples of Units

Multiple Prefix Symbol 1012 tera T 109 106 103

giga mega kilo

G M k

10-3 milli m 10-6 10-9

micro nano

JL n

10- 12 pico p

one hundredth of a meter, the common dimensions and multiples of which are found in Table 1-2. In the microscopic range, lengths are expressed as microme

ters (J,Lm),. nanometers (nm), and angstroms, A, some

times written A. Units are often multiplied by positive and negative powers of 10 to indicate their magnitude, the micrometer being 1 x 10-3 millimeters or 10- 4 em, the nanometer 0.001 um, and the angstrom 0.1 nm or 10-8 em, Although the micrometer (JLm) is the pre

ferred term for 0.001 mm in modern textbooks on colloid chemistry, the practice is sometimes to use the older and more familiar term, micron (JL). Similarly, the nanometer has replaced the millimicron (mJL). The student should be familiar with the prefixes (see Table 1-2) accompanying units such as mass, volume, and time. For example, a nanosecond, or ns, is 10- 9 second; a megaton (Mton) is 106 tons. Area is the square of a length and has the unit of square centimeters (sq. em or cm''), Volume. The measurable quantity, volume, is also

derived from length. Its reference standard is the cubic meter; its cgs unit is one millionth of this value or 1 cubic centimeter (cc or em"). Volume was originally defined in terms of the liter, the volume of a kilogram of water at 1 atmosphere pressure and 40 C, and was

meant to be equivalent to 1000em", Owing to the failure to correct for the dissolved air in the water, however, the two units do not compare exactly. It has since been established that 1 liter actually equals 1000.027 em". Thus, there is a discrepancy between the milliliter (one thousandth of a liter) and the cubic centimeter, but it is so slight as to be disregarded in general chemical and pharmaceutical practice. Volumes are usually expressed in milliliters in this book, abbreviated ml or mL, in conformity with the U.S. Pharmacopeia and the National 'Formulary; however, cubic centimeters are used in the book where this notation seems more appropriate.

The pharmacist uses cylindric and conical graduates, droppers, pipettes, and burettes for the measurement of volume; graduates are used more frequently than the other measuring apparatus in the pharmacy laboratory. The flared conical graduate is less accurate than the cylindric type, and the. use of the flared graduate should bediscouraged except for some liquids that need not be measured accurately. The selection of. the correct graduate for the volume of liquid to be measured has peen determined by Goldstein et ale 1

-I

Mass. The standard of mass is the kilogram. It is the mass of a platinum-iridium block preserved at the Bureau of Weights and Measures. The practical unit of mass in the cgs system is the gram (g), which is one thousandth of a kilogram. Mass is often expressed as the weight of a body. The balance is said to be used for "weighing," and the standard masses are known as "weights." The proper relationship between mass and weight will be considered under the topic of force. To weigh drugs precisely and accurately, the phar

macist must understand the errors inherent in operating a balance. A Class A balance, used for the compounding of prescriptions, is serviceable only if kept in good working condition and checked periodically for equality of arm length, beam rider accuracy, and sensitivity. These tests are described in the booklet by Goldstein and Mattocks. 2 Furthermore, a good balance is of no use unless an accurate set of weights is available. Density and Specific Gravity. The pharmacist fre

quently uses these measurable quantities when interconverting between mass and volume. Density is a derived quantity since it combines the units of mass and volume. It is defined as mass per unit volume at a fixed temperature and pressure and is expressed in the cgs system in grams per cubic centimeter (g/cm"), In SI units, density is expressed as kilograms per cubic meter. Specific gravity, unlike density, is a pure number

without dimension, however, it may be converted to density by the use of appropriate formulas." Specific gravity is defined as the ratio of the density of a substance to the density of water, the values for both substances being determined at the same temperature unless otherwise specified. The term specific gravity, in light of its definition, is a poor one; it would be more appropriate to refer to it as relative density. Specific gravity is defined more often for practical

purposes as the ratio of the mass of a substance to the mass of an equal volume of water at 4 or at some other specified temperature. The following notations are frequently found to accompany specific gravity readings: 25/25, 25/4, and 4/4. The first figure refers to

TABLE 1-3. Derived Oimensions andUnits

Chapter 1 Introduction 3

the temperature of the air in which the substance was weighed; the figure following the slash is the temperature of the water used. The official pharmaceutical compendia use a basis of 25/25 to express specific gravity. / Specific gravity may be determined by the use of various types of pycnometers, the Mohr-Westphal balance, hydrometers, and other devices. The measurements and calculations are discussed in elementary chemistry, physics, and pharmacy books.

Other Dimensions and Units. The derived dimensions and their cgs and SI units are listed in Table 1-3. Although the units and relations are self-explanatory for most of the derived dimensions, force, pressure, and energy require some elaboration. Force. One is familiar with force in everyday

experience as a push or pull required to set a body i.i motion. The larger the mass of the body and the greater the required acceleration, the greater the force that one must exert. Hence, the force is directly proportional to the lI1aSS (when acceleration is constant) and to the acceleration (when the mass is constant). This may be represented by the relation

Force ex Mass x Acceleration (1-1) This proportionality is converted to an equality, that is, to an equation or mathematical expression involving an equal sign, according to the laws of algebra, by the introduction of a constant. Accordingly, we write

f = k x m x a . (1-2) in whichfis the force, k is the proportionality constant, m is the mass, and a is the acceleration. If the units are chosen so that the constant becomes unity (i.e., has the value of 1), the well-known force equation of physics is obtained:

f= m x a (1-3) The cgs unit of force is the dyne, defined as the force that imparts to a mass of 1 g an acceleration of 1 cm/sec'', The reader should recall from physics that weight is

the force of gravitational attraction that the earth

Derived Dimensional Relationsh ip to Dimensions Symbol CGS Unit SI Unit OtherDimensions Area (A) Volume (V)

L2 L3

cm2 cm3 .

m2

m3 * the square of a length the cube of a length

Density (p) ML- 3 glcm3 kg m- 3 mass/unit volume Velocity (v) Acceleration (a) Force (f) Pressure (p) Energy (E)

LT- 1 LT- 2 MLT- 2 ML-lr-2 ML2T- 2

em/sec em/sec" g ern/sec" or dyne dyne/cm2 g cm2/sec2 or erg

m S-1 m S-2 kg m s- 2 = J m-1 = N N m- 2 = kg m-1s-2 = Pa t

4 Physical Pharmacy

exerts on a body, and it should be expressed properly in force units (dynes) rather than mass units (grams). The relationship between weight and mass can be obtained from equation (1-3). Substituting weight w for force and g for acceleration, the equation becomes

w = m x g (1-4) Although the gravitational acceleration of a body varies from one part of the earth to another, it is approximately constant at 981 cm/see'', Substituting this value for g, the weight of a 1-g mass is calculated from equation (1-4) as follows:

w = 1 g x 981 cm/sec'' and

w = 981 g cm/sec'' or 981 dynes Therefore, the weight of a body with a mass of 1 gram is actually 981 dynes. It is common practice to express weight in the mass unit, grams, since weight is directly proportional to mass; however, in problems involving these physical quantities, the distinction must be made. The SI unit of force is the newton (N), which is equal

to one kg m S-2. It is defined as the force that imparts to a mass of 1 kg an acceleration of 1 m/sec" (see Table 1-3). Pressure. Pressure may be defined as force per unit

area; the unit commonly used in science is dyne/em'', Pressure is often given in atmospheres (atm) or in centimeters or millimeters of mercury. This latter unit ' is derived from a measurement of the height of a column of mercury in a barometer, which is used. to measure the atmospheric pressure. The equation from elementary physics used .to convert height in a column of mercury or another liquid into pressure units is

pressure (dyne/ern') = p x g x h (1-5) where p is the density of the liquid in g/cm" at a particular temperature, g is the acceleration of gravity 980.665 cm/sec'', and h is the height in em of the column of liquid. At sea level, the mean pressure of the atmosphere supports a column of mercury 76 em (760 mm) or 29.9 inches in height. The barometric pressure may be translated into the fundamental pressure unit, dyne/cm'', by multiplying the height, h = 76 em, times 1 cm~ cross-sectional area by the density p of mercury, 13.595g/cm", at 00 to give the mass and multiplying this by the acceleration of gravity, g = 980.7 cm/sec'', The result divided by cm2 is 1.0133 x 106 dyne/ern" and is equal to 1 atm. This series of multiplication and division is expressed simply by equation 5. . In the SI system, the unit of pressure (or stress) is the newton divided by the meter squared (Nm-2) and is called the pascal (Pa), (see Table 1-3). Example 1- 1. Convert the pressure of a column of ethyl alcohol 76

em Hg (760nun Hg) high to a pressure at sea level and 0C (standard pressure) expressed in (a) dyne/em" and (b) pascals (Pa). The density (P) of ethanol at 0 Cis. 0.80625 g/cm",

(a) To obtain the standard pressure in dyne/em", one uses equation (1-5) with the density p =0.80625 g/em", the acceleration of gravity 9 at sea level as 980.665 em/see", and the height h of the column of mercury as 76.000 cm Hg.

Pressure = 0.80625 g/cm" x 980.665 em/sec" x 76.000 em = 6.00902 x 104 dyne/em''

(b) To obtain the standard pressure in pascals (Pa), we use SI units in equation (1-5):

kg (1Q2)3cm3)* (Pressure = 0.80625 g/cm" x - x --\ 103g 1m3

m x (980.665 cm/see2 x ~) x (76.000 em x 100 )100 em cm

2= 6.00902 x 103 kg m" . S-2 (or N . m- , or Pa)

*1 meter = lQ2em; therefore, 1 m3 = (102)3em3 = 106cm3. Work and Energy. Energy is frequently defined as the

condition of a body that gives it the capacity for doing work. The concept actually is so fundamental that no adequate definition can be given. Energy may be classified as kinetic energy or potential energy. The idea. of energy is best approached by way of the

mechanical equivalent of energy known as work and the therinal equivalent of energy or heat. When a constant force is applied to a body in the direction of its movement, the work done on the body equals the force multiplied by the displacement, and the system undergoes an increase in energy. The product of force and distance has the same dimensions as energy, namely ML2T--2 Other products also having the dimensions of energy are pressure x volume, surface tension x area, mass x velocity'', and electric potential difference x quantity of electricity. The cgs unit of work, also the unit of kinetic and

potential energy, is the erg. It is defined as the work done when a force of 1 dyne acts through a distance of 1 centimeter:

1 erg = 1 dyne x 1 em

The erg is often too small for practical use and is replaced by the joule (J) (pronounced jewel), which is equal to 107 ergs:

1 joule = 1 x 107 erg In carrying out calculations in the cgs system involving work and pressure, work must be expressed in ergs and pressure in dynes/em'', When using the 81 or any other system, consistent units mustalso be employed. Heat and work are equivalent forms of energy and

are interchangeable under certain circumstances. The thermal unit of energy in the cgs system is the gram calorie (small calorie). Formerly it was expressed as the amount of heat necessary to raise the temperature of 1 gram of water from 15 to 16 C. The small calorie is now defined as equal to 4.184 joules. The large or kilogram calorie (kcal) equals 1000 small calories. The SI unit for energy or work is the joule (J), which is seen in Table r-3 to be equivalent to the newton x meter (N m).

Chapter 1 Introduction 5

Temperature. Temperature is assigned a unit known as the degree. On the centigrade and the Kelvin or absolute scales, the freezing and boiling points of pure water at 1 atm pressure are separated by 100 degrees. . Zero degree on the centigrade scale equals 273.15 on the Kelvin scale.

SOME ELEMENTS OF MATHEMATICS . The student should become familiar with the funda

mental concepts of mathematics that are frequently used in the physical sciences and upon which are based many of the equations and graphic representations encountered in this book. Calculations Involving dimensions. Ratio and propor

tions are frequently used in the physical sciences for conversions from one system to another. The following calculation illustrates the use of proportions. Example 1-2. How many gram calories are there in 3.00 joules?

One should first recall a relationship or ratio that connects calories and joules. The relation 1 cal = 4.184 joules comes to mind. The question is then asked in the form of a proportion: If 1 calorie equals 4.184 joules, how many calories are there in 3.00 joules? The proportion is set down, being careful to express each quantity in its proper units. For the unknown quantity, aq "X" is used.

\ 1 cal X

4.184 joules 3.00 joules X = 3.00 joules x 1 cal

4.184 joules X = 0.717 cal

A second method, based on the requirementthat the units as well as the dimensions be identical on both sides of the equal sign, is sometimes more convenient than the method of proportions. Example 1-3. How many gallons are equivalent to 2.0 liters? It

would be necessary to set up successive proportions to solve this problem. In the method involving identity ofunits on both sides of the equation, the quantity desired, X (gallons), is placed on the left and its equivalent, 2.0 liters, is set down on the right side of the equation. The right side must then be multiplied by known relations in ratio form, such as 1 pint per 473 ml, to give the units of gallons. Carrying out. the indicated operations yields the result with its proper units.

X (ingallons) = 2.0 liter x (1000 mLlliter) x (1 ptl473mL) x (1 gaV8 pt)

X = 0.53 gal

One may be concerned about the apparent disregard for the rules of significant figures (p. 11) in the equivalents such as 1 pint = 473 mL. The quantity of pints. can be measured as accurately as that of milliliters, so that we assume 1.00 pint is meant here. The quantities 1 gallon and 1 liter are also exact by definition, and significant figures need not be considered in such cases.

Exponents. The various operations involving exponents, that is, the powers to which a number is raised, are best reviewed by studying the examples set out in Table 1-4. .

TABLE 1-4. The Rules DfEXpDnents

a x a x a = a3

a2 .x a3 = a2 +3 = as

---

I .

6 Physical Pharmacu

and taking the natural logarithm, equation (1-12) becomes

In a = In lOX = x In 10 (1-13) Now In 10 = 2.303, and equation (1-13) becomes

In a = 2.303 x (1-14) Substituting the identity x = log a from equation (1-11) into equation (1-14) gives the desired formula. The application of logarithm is best demonstrated by

considering several examples. In the expression, log 60.0 = 1.778

the digit 1 to the left of the decimal point in the logarithm is known as the characteristic and signifies that the number 60.0 belongs to that class of numbers with a magnitude of 101 and thus contains two figures to the left of the decimal point. The quantity 0.778 of the logarithm is known as the mantissa and is found in the table of common logarithms. It is often convenient to express the number 60.0 by writing it with one significant figure to the left of the decimal point, 6.00, multiplied by 10 raised to the first power, viz., 6.00 x 101 The exponent of 10 then gives the characteristic, and the value in the logarithm table gives the mantissa directly. This method may be used to obtain the logarithm of

6000 from a table as follows. The number is first written as 6.000 x 103if it is accurate to four significant figures. The characteristic is observed to be 3, and the mantissa is found in the table as 0.778. Hence,

log 6000 = 3.778 For decimal fractions that frequently appear in problems involving molar concentration, the following method is used. Suppose one desires to know the logarithm of 0.0600. The number is first written as 6..00 x 10- 2 The characteristic of a number may be positive or negative; the mantissa is always positive. The characteristic in this case is -2 and the mantissa is 0.778. Hence,

log 0.0600 = -2 + 0.778 =.-1.222 Finding the number in a table when the logarithm is

given, that is, obaining the antilogarithm, is shown by the following example. What is the value of a if log a = 1.7404? The characteristic is 1 and the mantissa is 0.7404. From the table of logarithms, one finds that the number corresponding to a mantissa of 0.7404 is 5.50. The characteristic is 1, so the antilogarithm is 5.50 x 101 or 55.0. . Let us find the antilogarithm of a negative number,

-2.699. Recalling that the mantissa must always be positive, we first separate the logarithm into a negativecharacteristic and positive mantissa:

-2.699 = -3.00 + 0.301 This transformation is easily seen in Figure 1-1, in

00

-1-1

+-3

-2 1 -2 45' 0.301-3 -3

0

-1~l -2.699

-2- 1 -3-~

Fil. 1- 1. Schematic representation for finding the antilogarithm of a negative number.

which -2.699 corresponds to going down the scale in a negative direction to -3 and coming back up the scale 0.301 units in the positive direction. Actually, by this process, we are subtracting 1 from the c~aracteristic and adding 1 to the mantissa, or to the quantity

-2.699 = (-2) + (-0.699) we subtract and add 1 to yield

(-2 - 1) + (-0.699 + 1) = -3 + 0.301 The result (-3 + 0.301) is sometimes abbreviated to (3.301), in which the minus sign above the 3 applies only to thecharacteristic. 3" is commonly referred to as "bar three." It has been the practice in some fields, such as quantitative analysis, to use the form in which 1~ is added and subtracted to give

3.301 = 7.301 - 10 For physical chemical calculations and for plotting

logarithms of numbers, it is more convenient to use the form -2.699 than one of the forma having a mixture of negative and positive parts. For use with logarithm tables, however, the mixed form is needed. Thus, in order to obtain the antilogarithm, we write the logarithm as 3.301. The number corresponding to the mantissa is found in the logarithm table to be 2.00. The characteristic is observed to be -a, and the final result is therefore 2.00 x 10-3. We have dealt with logarithms to the base 10

(common logarithms) and to the base e = 2.71828. (natural logarithms). Logarithms to any other positive number as the base, b, may also be obtained. The formula used for this purpose is

logb(a) = loge(a)lloge(b) (1-15) To obtain the logarithm of the number a = 100 to the base b = 37, we substitute in equation (1-15):

log37(100) = In(100)lln(37) = 1.2753 Onemay also use commonlogs on the right side of the

equation: log37(lOO) = loglo(100)lloglo(37) = 1.2753

7

TABLE 1-5. Rules of Logarithms logab = loga + log b 1log- = log 1 - loga = -log a

a

log~ = loga - log b log a2 = loga + log a = 2 log a log 1 = 0 since 10 = 1 logva = loga1l2 = ~ loga

loga- 2 = - 2 loga = 2 log~ a

These formulas allow one to obtain a logarithm to a base b for any whole or fractional positive number desired. Problem 1-12 is an exercise involving the change from one logarithmic base to another. As seen in the table of exponents (Table 1-4),

numbers may be multiplied and divided by adding and subtracting exponents. Since logarithms are exponents, they follow the same rules. Some of the properties of logarithms are exemplified by the identities collected in Table 1-5. Variation. The scientist is continually attempting to

relate phenomena and establish generalizations with which to consolidate and interpret experimental data. The problem frequently resolves itself into a search for ~he relationship between two quantities that are changmg at a certain rate or in a particular manner. The dependence of one property, the dependent variable y, on the change or alteration of another measurable quantity, the independent variable x, is expressed mathematically as

y ex X (1-16) which is read: "y varies directly as x," or "y is directly proportional to z." A proportionality is changed to an equation a~ follows. If y is proportional to x in general, then all pairs of specific values of y and s, say YI and Xl' Y2 and X2, , are proportional. Thus

Yl Y2-:=-= .... (1-17) Xl X2

Since the ratio ?f any y to its corresponding X is equal ~o any other ratio of y and x, the ratios are constant, or, In general

1- = const~nt (1-18)x

Hence, it is ~ simpl~ matter to change a proportionality to an equality by Introducing a proportionality constant, k. To summarize, if

TABLE 1- 6. Formulas "1IIustratin, thePrinciple of Variation

Chapter 1 Introduction

yexx

then y = kx (1-19)

It is frequently desirable to show the relationship between X and y by the use of the more general notation,

y = fix) (1-20)

which is read: "y is some function of x." That is, y may be e~ual to 2x, .to 27x2, or to 0.0051 + log (a/x). The functional notation, equation (1-20), merely signifies that y and x are related in some way without specifying the actual equation by which they are connected. Some well-known formulas illustrating the principle of variation are shown in Table 1-6.

Graphi~ Methods. Scientists are not usually so fortunate as to begin each problem with an equation at hand r~lating the variables under study. Instead, the investigator must collect raw data and put them in the form of a table ?r graph to b~tter observe the relationships. Constructing a graph WIth the data plotted in a manner

~o as. to form a smooth curve often permits the investigator to observe the relationship more clearly, and perhaps allows expression of the connection in the form of a mathematical equation. The procedure of obtaining an empiric equation from a plot of the data is known as curve fitting and is treated in books on statistics and graphic analysis. The magnitude of the independent variable Iscustom

arily measured along the horizontal coordinate scale called the X axis. The dependent variable is measured along the vertical scale or the y axis. The data are plotted on the graph, and a smooth line is drawn through the points. The X value of each point is known as the X coordinate or the abscissa, the y value is known as the y coordinate or the ordinate. The intersection of the X axis and the y axis is referred to as the origin. The X and y values may be either negative or positive. The simplest relationship between two variables

where the variables contain no exponents other than one (first-degree equation), yields a straight line when plotted on rectangular graph paper. The straight-line or linear relationship is expressed as

y = a + b (1-21) in which y is the dependent variable, x is the independent variable, and a and b are constants. The constant

Dependent Independent ProportionaIity Measurement Equation Variation Variable Constant Circumference of a circle C 'frD Circumference, C Diameter, D 'fr = 3.14159 . Density M = pV Mass, M Volume, V Density,p Distance of falling body s = !gt2 Distance, S Time, t2 Gravity constant, !gC .Freezing point depression 41, ~ K,m Freezing point depression, 4Tf Molality, m ryOSCOPIC constant, K, 2

8 Physical Pharmacu

b is the slope of the line; the greater the value of b, the steeper the slope. It is expressed as the change in y with the change in x or b = ~; b is also the tangent of the. angle that the line makes with the x axis. The slope may be positive or negative depending on whether the line slants upward or downward to the right. When b = 1, the line makes an angle of 45with the x axis (tan 45 = 1), and the equation of the line may then be written

y = a + x (1-22) When b = 0, the line is horizontal (i.e., parallel to the x axis), and the equation reduces to

y=a (1-23) The constant a is known as the y intercept and

signifies the point at which the line crosses the y axis. If a is positive, the line crosses the y axis above the x axis; if negative, it intersects the y axis below the x axis. When a is zero, equation (1-21) may be written,

y = bx (1-24) and the line passes through the origin. The results of the determination of the refractive

index of a benzene solution containing increasing concentrations of carbon tetrachloride are found in Table 1-7. The data are plotted in Figure 1-2 and are seen to produce a straight line with a negative slope. The equation of the line may be obtained by using the two-point form of the linear equation,

Y2 - YI y - YI = (x - Xl) (1-25)X2 - Xl

The method involves selecting two widely separated points (Xl' YI) and (X2' Y2) on the line and substituting into the two-point equation. Ex.mpl, 1-4. Referring to Figure 1-2, let 10.0% be Xl and its

corresponding y value 1.497 be YI; let 60.0% be X2 and 1.477 be Y2. The equation then becomes

y - 1.497 = 1~~~ =:O~:7 (3: - 10.0) Y - 1.497 = -4.00 x 10-4 (X -.10.0)

Y = -4.00 X 10-4 X + 1.501

The value -4.00 x 10-4 is the slope of the straight line and corresponds to b in equation (1-21). A negative

TABLE 1-7. Refractive Indices of Mixtures of Benzen, and Carbon Tetrachloride (x) (y) Concentration of CCI 4 Refractive (volume %) Index 10.0 1.497 25.0 1.491 33.0 1.488 50.0 1.481 60.0 1.477

y intercept = 1.501 ,1.500 ,

, XhYl

~SIOpe = -4.00 X 10-4

)(

~ 1.490 .5

; : o C'G ~ CDa::

1.480 X~,Y2

Equation of line ,Y = -4.00 X 10- 4x + 1.501 ,,

,

20 40 60 Carb~n tetrachloride, % by volume

Fig. 1-2. Refractive index of the system benzene-carbon tetrachloride.

value for b indicates that y decreases with increasing values of x as observed in Figure 1-2. The value 1.501 is the y intercept and corresponds to a in equation (1-21).* It may be obtained from the plot in Figure 1-2 by extrapolating (extending) the line upwards to the left until it intersects the yaxis. It will also be observed that

Y2 - Yl = fiy = b (1-26)X2' - Xl fix

and this simple formula allows one to compute the slope of a straight line. The use of statistics to determine whether data fit the slope of such a line and its intercept on the y axis is illustrated later in this chapter. Not all experimental data form straight lines. when

plotted on ordinary rectangular coordinate paper. Equations containing x2 or y2 are known as seconddegree or quadratic equations, and. graphs of these equations yield parabolas, hyperbolas, ellipses, and circles. The graphs and their corresponding equations may be found in standard textbooks on analytic geometry. Logarithmic relationships occur frequently in scien

tific work. The data relating the amount .ofoil separating from an emulsion per month (dependent variable, y) as a function of the emulsifier concentration (independent variable, ~) are collected in Table 1-8. The data from this experiment may be plotted in

several ways. In Figure 1-3, the oil separation y is plotted as ordinates against the emulsifier concentration x as abscissas on a rectangular coordinate grid. In

*The y-intercept, 1.501, is of course the refractive index of pure benzene at 20C. For the purpose of this example weassume that we were not able to find this value in a table of refractive indices. In handbooks of chemistry and physics the value is actually found to be 1.5011 at 20C.

Chapter 1 Introduction 9

TABLE 1-8. Emulsion Stability asa Function of Emulsifier Concentration

Emulsifier Oil Separation Logarithm of (% Concentration) (mUmonth) Oil Separation (x) (y) (logy) 0.50 5.10 0.708 1.00 3.60 0.556 1.50 2.60 0.415 2.00 2.00 0.301 2.50 1.40 0.146 3.00 1.00 0.000

Figure 1-4, the logarithm of the oil separation is plotted against the concentration. In Figure 1-5, the data are plotted on semilogarithm paper, consisting of a logarithmic scale on the vertical axis and a linear scale on the horizontal axis.

Although Figure 1-3 provides a direct reading of oil separation, difficulties arise when one attempts to draw a smooth line through the points or to extrapolate the curve beyond the experimental data. Furthermore, the equation for the curve cannot be obtained readily from Figure 1-3. When the logarithm of oil separation is plotted as the ordinate, as in Figure 1-4, a straight line results, indicating that the phenomenon follows a logarithmic or exponential relationship. The slope and the y intercept are obtained from the graph, and the equation for the line is subsequently found by use of the two-point formula:

log y = 0.85 - 0.28x Figure 1-4 requires that we obtai-i the logarithms of the oil-separation data before the graph is constructed and, conversely, that we obtain the antilogarithm of the ordinates to read oil separation from the graph. These inconveniences of converting to logarithms and antilogarithms may be overcome by the use of semilogarithm paper. The x and y values of Table 1-8 are plotted directly on the graph to yield a straight line, as seen in

6.0

i ~

E

10 Physical Pharmacu

with numbers much like desk and hand-held calculators do. The modern calculator is provided with registers for the storage of data and a central core that can be programmed with mathematical instructions to carry out most mathematical functions. The programmable calculator, like the computer, also has a decisionmaking capacity through its ability to determine whether a number is larger than zero (positive), smaller than zero (negative), or equal to zero. The computer differs from the calculator in that it is faster, capable of greater storage, and more versatile.

The analog computer, unlike the digital computer, handles mathematical problems using voltages to represent variables such as concentration, pressure, time, and temperature. If the problem can be written as' a differential equation (Examples A-9, A-12, p. 599), the solution is obtained by expressing the equation in voltages, capacitances., and resistances and then causing the voltage to vary with time as determined by the differential equation. The solution to the problem appears as a graphic plot on a recorder or is displayed as a tracing on an oscilloscope screen. The analog computer is composed of tens or hundreds of amplifiers that are used for the mathematical operations of addition, multiplication, and so on. The amplifiers are connected by the operator into a circuit that represents the differential equation at hand. Each amplifier corresponds to one step in the chemical, physical, or mechanical process under consideration. The analog computer is used in engineering to simulate the spring action on the axles of an automobile or the movement of a skyscraper in a high wind. It has been used to calculate the absorption, distribution, and elimination constants for a drug that is administered to a patient and to plot the curves for uptake and excretion. Today the digital computer can also calculate such values and prepare graphs with facility, and the popularity of the analog computer has diminished in pharmacy in recent years.

Currently, the microcomputer and the hand-held calculator are of great interest in small business and education and for personal use. The microcomputer, at a price within reach of the average individual, is more powerful today than the large institutional computers of the 1960s.

Programs may be written for large electronic computers and microcomputers in a number of languages, the most popular of which are FORTRAN, BASIC, PASCAL, C, and C++. Even some hand-held calculators can now be programmed in BASIC.

It will profit the student to become familiar with BASIC and/or FORTRAN and with the operation of an institutional or personal microcomputer. A hand-held calculator will be useful for working the problems atthe ends of the chapters of this book. A programmable calculator is particularly convenient to obtain the slopes and intercepts of lines and for carrying out a repetitive sequence of mathematical operations. For example, in the chapter on solubility, if one desires to calculate the

minimum pH for complete solubility of a series of 10 acidic drugs of known pKa values, it is simpler and faster to program the calculator than to carry out a number of repetitive steps for each of the 10 drugs.

Significant Figures. A significant figure is any digit used to represent a magnitude or quantity in the place in which it stands. The number zero is considered as a significant figure except when it is used merely to locate the decimal point. The two zeros immediately following the decimal point in the number 0.00750 merely locate the decimal point and are not significant. However, the zero following the 5 is significant since it is not needed to write the number; if it were not significant, it could be omitted. Thus, the value contains three significant figures. The question of significant figures in the number 7500.is ambiguous. One does not know whether any or all of the zeros are meant to be significant, or whether they are simply used to indicate the magnitude of the number. To express the significant figures of such a value in an unambiguous way, it is best to use exponential notation. Thus, the expression 7.5 x lOa signifies that the number contains two significant figures, and the zeros in 7500 are not to be taken as significant. In the value, 7.500 x 103, both zeros are significant, and the number contains a total of four significant figures. The significant figures of some values are shown in Table 1-9.

The significant figures of a number include all certain digits plus the first uncertain digit. For example, one may use a ruler, the smallest subdivisions of which are centimeters, to measure the length of a piece of glass tubing. If one finds that the tubing measures slightly greater than 27 em in length, it is proper to estimate the doubtful fraction, say 0.4, and express the number as 27.4 em. A replicate measurement may yield the value 27.6 or 27.2 em so that the result is expressed as 27.4 0.2 em. When a value such as 27.4 em is encountered in the literature without further qualification, the reader should assume that the final figure is correct to within about 1 in the last decimal place, which is meant to signify the mean deviation of a single measurement. However, when a statement is given in the official compendia (U.S. Pharmacopeia and National Formulary) such as "not less than 99," it means 99.0 and not 98.9.

Significant figures are particularly useful for indicating the precision of a result. The precision is limited by

TABLE 1-9. Sifnificant Figures Number of

Number Significant Figures

53. 2 530.0 4

0.00053 2 5.0030 5 5.3 x 10-2 2 5.30 X 10-4 3

53000 indeterminate

the instrument used to make the measurement. A measuring rule marked off in centimeter divisions will not produce as great a precision as one marked off in 0.1 em or mm. One may obtain a length of 27.4 0.2 ern with the first ruler and a value of, say, 27.46 0.02 em with the second. The latter ruler, yielding a result with four significant figures, is obviously the more precise one. The number 27.46 implies a precision of about 2 parts in 3000, whereas 27.4 implies a precision of only 2 parts in 300. The absolute magnitude of a value should not be

confused with its precision. We consider the number 0.00053 molelliter as a relatively small quantity because three zeros immediately follow the decimal point. But these zeros are not significant and tell us nothing about the precision of the measurement. When such a result is expressed as 5.3 x 10-4 molelliter, or better as 5.3 (O.l) x 10-4 molelliter, both its precision and its magnitude are readily apparent. In dealing with experimental data, certain rules

pertain to the figures that enter into the comp.itations: 1. In rejecting superfluous figures, increase by 1 the

last figure retained if the following figure rejected is 5 or greater. Do not alter the last figure if the rejected figure has a value of less than 5. Thus, if the value 13.2764 is to be rounded off to four significant figures, it is written as 13.28. The value 13.2744 is rounded off to 13.27. 2. In addition or subtraction, include only as many

figures to the right of the decimal point as there are present in the number with the least such figures. Thus, in adding 442.78, 58.4, and 2.684, obtain the sum and then round off the result so that it contains only one figure following the decimal point:

442.78 + 58.4 + 2.684 = 503.684 This figure is rounded off to 503.9. Rule 2 of course cannot apply to the weights and

.volumes of ingredients in the monograph of a pharmaceutical preparation. The minimum weight or volume of each ingredient in a pharmaceutical formula or a prescription should be large enough that the error introduced is no greater than, say, 5 in 100 (5%), using the weighing and measuring apparatus at hand. Accuracy and precision in prescription compounding is discussed in some detail by Brecht. 5 3. In multiplication or division, the rule commonly

used is to retain the same number of significant figures in the result as appear in the value with the least number of significant figures. In multiplying 2.67 and 3.2, the result is recorded as 8.5 rather than as 8.544. A better rule here is to retain in the result the number of figures that produces a percentage error no greater than that in the value with the largest percentage uncertainty. 4. In the use of logarithms for multiplication and

division, retain the .same number of significant figures in the mantissa as there are in the original numbers.

Chapter 1 Introduction 11

The characteristic signifies only the magnitude of the number and accordingly is not significant. Since calculations involved in theoretic pharmacy usually require no more than three significant figures, a four-place logarithm table yields sufficient precision for our work. Such a table is found on the inside back cover of this book. The hand calculator is more convenient, however, and tables of logarithms are used less frequently today. Logarithms to nine significant figures are obtained by the simple press of a button on the modern hand calculator. 5. If the result is to be used in further calculations,

retain at least one digit more than suggested in the rules just given. The final result is then rounded off to the last significant figure.

STATISTICAL METHODS AND THE ANALYSIS OF ERRORS If one is to maintain a high degree of exactitude in the

compounding of prescriptions and the manufacture of products on a large scale, one must know how to locate and eliminate constant and accidental errors insofar as possible. Pharmacists must recognize, however, that just as they cannot hope to produce a perfect pharmaceutical product, neither can they make an absolute measurement. In addition to the inescapable imperfections in mechanical apparatus and the slight impurities that are always present in chemicals, perfect accuracy is impossible because of the inability of the operator to make a measurement or estimate a quantity to a degree finer than the smallest division of the instrument scale. Error may be defined as a deviation from the absolute

value or from the true average of a large number of results. Two types of errors are recognized: determinate (constant) and indeterminate-(random or accidental) . Determinate Errors. Determinate or constant errors are

those that, although sometimes unsuspected, may be avoided or determined and corrected once they are uncovered. They are usually present in each measurement and affect all observations of a series in the same way. Examples of determinate errors are those inherent in the particular method used, errors in the calibration and the operation of the measuring instruments, impurities in the reagents and drugs, biased personal errors that, for example, might recur consistently in the reading of a meniscus, in pouring and mixing, in weighing operations, in matching colors, and in making calculations. The change of volume of solutions .with temperature, while not constant, is, however, a systematic error that also may be determined and accounted for once the coefficient of expansion is known. Determinate errors may be combated in analytic

work by the use of calibrated apparatus, by the use of blanks and controls, by using several different analytic

12 Physical Pharmacy

procedures and apparatus, by eliminating impurities, and by carrying out the experiment under varying conditions. In pharmaceutical manufacturing, determinate errors may be eliminated by calibrating the weights and other apparatus and by checking calculations and results with other workers. Adequate corrections' for determinate errors must be made before the estimation of indeterminate errors can have any significance. Indeterminate Errors. Indeterminate errors occur by

accident or chance, and they vary from one measurement to the next. When one fires a number of bullets at a target, some may hit the "bull's eye," while others will be scattered around this central point. The greater the skill of the marksman, the less scattered will be the pattern on the target. Likewise, in a chemical analysis, the results of a series of tests will yield a random pattern around an average or central value, known as the mean. Random errors will also occur in filling a number of capsules with a drug, and the finished products will show a definite variation in weight. Indeterminate errors cannot be allowed for or cor

rected because of the natural fluctuations that' occur in all measurements. Those errors that arise from random fluctuations in

temperature or other external factors and from the variations involved in reading instruments are not to be considered accidental or random. Instead, they belong to the class of determinate errors and are often called pseudoaccidental or variable determinate errors. These errors may be reduced by controlling conditions through the use of constant-temperature baths and ovens, the use of buffers, and the maintenance of constant humidity and pressure where indicated. Care in reading fractions of units on graduates, balances, and other apparatus can also reduce pseudoaccidental errors. Variable determinate errors, although seemingly indeterminate, can thus be determined and corrected by careful analysis and refinement of technique on the part of the worker. Only errors that result from pure random fluctuations in nature are considered truly indeterminate. Precision and Accuracy. Precision is a measure of the

agreement among the values in a group of data, while accuracy is the agreement between the data and the true value. Indeterminate or chance errors influence the precision of the results, and the measurement of the precision is accomplished best by statistical means. Determinate or constant errors affect the accuracy of data. The techniques used in analyzing the precision of results, which in turn supply a measure of the indeterminate errors, will be considered first, and the detection and elimination of determinate errors or inaccuracies will be discussed later. Indeterminate or chance errors obey the laws of

probability, both positive" and negative errors being equally probable, and larger errors being less probable than smaller ones. If one plots a large number of results

~--99.7%----+-l 95.5%

vg%J

- 30 - 20 - 0 0 +0 +20 +30 Jl~i

7 8 9 10 11 12 13 14 15 16 17

Fig. 1-6. The normal curve for the distribution of indeterminate errors.

having various errors along the vertical axis against the magnitude of the errors on the horizontal axis, a bell-shaped curve, known as a normal frequency distribution curve, is often obtained, as shown in Figure 1-6. If the distribution of results follows the normal probability law, the deviations will be represented exactly by the curve for an infinite number of observations, which constitute the universe or population. Whereas the population is the whole of the category under consideration, the sample is that portion of the population used in the analysis.

The Arithmetic Mean. When a normal distribution is obtained, it follows that .the arithmetic mean is the best measure of the central value of a distribution; that is, the mean represents a point corresponding closest to the "bull's eye." The theoretic mean for a large number of measurements (the universe or population) is known as the universe or population mean and is given the symbol ..., (mu). The arithmetic mean X is obtained by adding to

gether the results of the various measurements and dividing the total by the number N of the measurements. In mathematical notation, the arithmetic mean for a small group of values is expressed as

X = I(Xi) (1-27)N in which I is the Greek capital letter sigma standing for "the sum of," Xi is the ith individual measurement of the group, and N is the number of values. X is an estimate of ..., and approaches it as the number of measurements N is increased, Measures of Dispersion. After having chosen the

arithmetic mean as the central tendency of the data, it is necessary to express the dispersion or scatter about the central value in a quantitative fashion so as to establish an estimate of variation among the results. This variability is usually expressed as the Ta1"ge, the mean deviation, or the standard deviation.

The range is the difference between the largest and the smallest value in a group of data and gives a rough idea of the.dispersion. It sometimes leads to ambiguous results, however, when the maximum and minimum values are notIn line with the rest of the data. The range will not be considered further. The average distance of all the hits from the "bull's

eye" would serve as a convenient measure of the scatter on the target. The average spread about the arithmetic mean of a large series of weighings or analyses is the mean deviation 8 of the population.* The sum of the positive and negative deviations about the mean equals zero; hence, the algebraic signs are disregarded in order to obtain a measure of the dispersion. The mean deviation d for a sample, that is, the

deviation of an individual observation from the arithmetic mean of the sample, is obtained by taking the difference between each individual value Xi and the arithmetic mean X, adding the differences without regard to the algebraic signs, and dividing the sum by the number of values to obtain the average. The mean deviation of a sample is expressed as

(1-28)

in which IIXi - XI is the sum of the absolute deviations from the mean. The vertical lines on either side of the term in the numerator indicate that the algebraic sign of the deviation should be disregarded. Youden" discourages the use of the mean deviation

since it gives a biased estimate that suggests a greater precision than actually exists when a small number of values are used in the computation.Furthermore, the mean deviation of small subsets may be widely scattered around the average of the estimates, and accordingly, d is not particularly efficient as a measure of precision. The standard deviation (J' (the Greek lower case letter

sigma) is the square root of the mean of the squares of the deviations. This parameter is used to measure the dispersion or variability of a large number of measurements; for example, the weights of the contents of several million capsules. This set of items or measurements approximates the population or the universe, and (J is therefore called the standard deviation of the universe. **Universe standard deviations are shown in Figure 1-6.

*The population mean deviation is written as IIXi - 11-1

& = N

where Xi is an individual measurement, II-is the population mean, and N is the number of measurements. **Theequation for the universe standard deviation is

IT = ' 100, one may use N instead of (N - 1) to estimate the population standard deviation, since the difference between the two is negligible. Modern statistical methods handle the small sample

quite well; however, the investigator should recognize that the estimate of the standard deviation becomes less reproducible and, on the average, becomes lower than the universe standard deviation as fewer samples are used to compute the estimate. A sample calculation involving the arithmetic mean,

the mean deviation, and the estimate of the standard deviation follows. Example '-5. A pharmacist received a prescription for a patient

with rheumatoid arthritis calling for seven divided powders, the contents of which were each to weigh 1.00gram. To check his skill in filling the powders, he removed the contents from each paper after filling the prescription by the block-and-divide method and then weighed the powders carefully. The results of the weighings are given in the first column of Table 1-10; the deviations of each value from the arithmetic mean, disregarding the sign, are given in column 2; and the squares of the deviations are shown in the last column. Based on the use of the mean deviation, the weight of the powders may be expressed as 0.98 ::t 0.046 gram. The variability of a single

14 Physical Pha171WCY

TABLE 1-10. StMiltical Analysis Df Divided PDwder Compoundin6 Technique

Weight of Deviation, Square of the Powder Contents (Sign -,gnored) Deviation (grams) IXi - XI (Xi - X)2

1.00 0.02 0.0004 0.98 0.00 0.0000 1.00 0.02 0.0004 1.05 0.07 0.0049 0.81 0.17 0.0289 0.98 0.00 0.0000 1.02 0.04 0.0016

Total I = 6.84 I = 0.32 I = 0.0362 Average 0.98 0.046

powder may. also be expressed in terms of percent deviation by dividing the mean deviation by the arithmetic mean and multiplying by 100. The result is 0.98 ::!: 4.6%; of course, it includes errors due to removing the powders from the papers and weighing the powders in the analysis.

The standard deviation is used more frequently than the mean deviation in research work. For large sets of data, it is approximately 25% larger than the mean deviation, that is, (J = 1.25 8. The statistician has estimated that owing to chance

errors, about 68% of all results in a large set will fall within one standard deviation on either side of the arithmetic mean, 95.5% within 2 standard deviations, and 99.7% within 3 standard deviations, as seen in Figure 1-6. Goldstein7 selected 1.73 (J as an equitable tolerance

standard for prescription products, whereas Saunders and Fleming'' advocated the use of 3 (J as approximate limits of error for a single result. In pharmaceutical work, it should be considered

permissible to accept 2 8 as a measure of the variability or "spread" of the data in small samples. Then, roughly 5 to 10% of the individual results will be expected to fall outside this range if only chance errors occur. The estimate of the standard deviation in Example

1-.1,. is calculated as follows:

~0.0362s = (7 _ 1) = 0.078 gram and 2 s is equal to 0.156 gram. That is to say, based upon the analysis of this experiment, the pharmacist should expect that roughly 90 to 95% of the sample values will fall within 0.156 gram of the sample mean. The smaller the standard deviation estimate (or the

mean deviation) the more precise is the compounding operation. In the filling of capsules, precision is a measure of the ability of the pharmacist to put the same amount of drug in each capsule and to reproduce the result in subsequent operations. Statistical techniques for predicting the probability of occurrence of a specific deviation in future operations, although important in

pharmacy, require methods that are outside the scope of this book. The interested reader is referred to treatises on statistical analysis. Whereas the average deviation and the standard

deviation can be used as measures of the precision of a method, the difference between the arithmetic mean and the true or absolute value expresses the error that can often be used as a measure of the accuracy of the method. The true or absolute value is ordinarily regarded as

the universe mean IJr- that is, the mean for an infinitely large set-since it is assumed that the true value is approached as the sample size becomes progressively larger. The universe mean does not, however, coincide with the true value of the quantity measured in those cases in which determinate errors are inherent in the measurements. The difference between the sample arithmetic mean

and the true value gives a measure of the accuracy ofan operation; it is known as the mean error. In Example 1-5, the true value is 1.00 gram, the

amount requested by the physician. The apparent error involved in compounding this prescription is

E = 1.0 - 0.98 = +0.02 gram in which the positive sign signifiesthat the true value is greater than the mean value. An analysis of these results. shows, however, that this difference is not statistically significant, but rather is most likely due to accidental errors.* Hence, the accuracyof the operation in Example 1-5 is sufficiently great that no systemic error can be presumed. We ~ay find on further analysis, however, that one or several results are questionable. This possibility is considered later. If the

*The deviation of the arithmetic mean from the true value or the mean of the parent population can be tested by use of the following expression:

X-ILt = """':';--'-8/~

In this equation, t is a statistic known as Student's t value, after W. S. Gosset, who wrote under the pseudonym of "Student." The other terms in the equation have the meaning previously assigned to them.

Student's t Values for Six Degrees of Freedom Probability of a plus or minus deviation greater than t

0.8 0.6 0.4 0.2 0.02 0.001 t value 0.27 0.55 0.91 1.44 3.14 5.96 Substituting the results of the analysis of the divided powders into

the equation just given, we have t = X - IL = 0.98 - 1.00 = -0.02 = -0.61

8/~ 0.08rv6 0.033 Entering the table with a t value of 0.61, we see that the probability of finding a deviation greater than t is roughly equal to 0.6. This means that in the long run there are about 60 out of 100 chances of finding a t value greater than -0.61 by chance alone. This probability is sufficiently large to suggest that the difference between the mean and the true value may be taken as due to chance.

Chapter 1 Introduction 15

arithmetic mean in Example 1-5 were 0.90 instead of 0.98 the difference could be stated with assurance to have statistical significance, since the probability that such a result could occur by chance alone would be small.* The mean error in this case is

1.00 - 0.90 = 0.10 gram The relative error is obtained by dividing the mean

error by the true value. It may be expressed as a percentage by multiplying by 100, or in parts per thousand by multiplying by 1000. It is easier to compare several sets of results by using the relative error rather than the absolute mean error. The relative error in the case just cited is

0.10 gram x 100 = 10% 1.00 gram

The reader should recognize that it is possible for a result to be precise without being accurate, that is, a c098tant error is present. If the capsule contents in Example 1-5 had yielded an average weight of 0.60 gram with a mean deviation of 0.5%, the results would have been accepted as precise. The degree of accuracy, however, would have been low since the average weight would have differed from the true value by 40%. Conversely, the fact that the result may be accurate does not necessarily mean that it is also precise. The situation can arise in which the mean value is close to the true value, but the scatter due to chance is large. Saunders and Fleming" observe that "it is better to be roughly accurate than precisely wrong." A study of the individual values of a set often thro.ws

additional light on the exactitude of the compounding operations. Returning to the data of E.xample 1-5, we note one rather discordant value, namely, 0.81 gram. If the arithmetic mean is recalculated ignoring this measurement, one obtains a mean of'1.01 grams. The mean deviation without the doubtful result was 0.02 gram. It is now seen that the divergent' result is 0.20 gram smaller than the new average or, in other words, its deviation is 10 times greater than the mean deviation. A deviation greater than four times the mean devi.ati~n will occur purely by chance only about once or twice In 1000 measurements; hence, the discrepancy in this case was probably caused by some definite error. in. ~echnique. This rule is rightly questione~ by sta.tIst~cIans, but it is a useful though not always reliable cntenon for finding discrepant results. \

*When the mean is 0.90 gram, the t value is

t = 0.90 -1.00 = -0.10"" -3 0.08/"\16 0.033

and the probability of finding a t value greater than -3 as a result ?f chance alone is found in the t table to be about 0.02, or 2 chances Itt 100. This probability is sufficiently small to suggest that the differencebetween the mean and the true value is real, and the error may be computed accordingly.

TABLE 1- 11. Refractive Indices of Mixtures of Benzene and Carbon Tetrachloride (x) Concentration (y) of CCI4 Refractive (volume %) Index 10.0 1.497 26.0 1.493 33.0 1.485 50.0 1.478 61.0 ,1.477

Having uncovered the variable weight among the units, one can proceed to investigate the cause of the determinate error. The pharmacist may find that some of the powder was left on the sides of the mortar or on the weighing paper or possibly was lost during trituration. If several of the powder weights deviated widely from the mean, a serious deficiency in the compounder's technique would be suspected. Such appraisals as these in the college laboratory Will aid the student in locating and correcting errors and will help the pharmacist to become a safe and proficient compounder before entering the practice of pharmacy. Binge~heimer~ has ?escribed such a program for students In the dispensing laboratory.

Linear Regression Analysis. The data given in Table 1-7 and plotted in Figure 1-2 clearly indicate ~he existence of a linear relationship between the refractive index and volume percent of carbon tetrachloride in benzene. The straight line that joins virtually all the points can be drawn readily on the figure by. sight~ng the points along the edge of a ruler and drawing a line that can be extrapolated to the y axis with confidence. Let us suppose, however, that the person who

prepared the solutions and carried out the refractive index measurements was not skilled and, as a result of poor technique, allowed indetermin~te errors to. appear. We might then be presented WIth the data given in Table 1-11. When this is plotted on graph paper, an appreciable scatter is observed (Fig. 1-7) and we ::-re unable with any degree of confidence, to draw the line that expresses .the relation between refractive index and concentration. It is here that we must employ better means of analyzing the available data. The first step is to determine whether or not the data

in Table 1-11 should fit a straight line, and for this we calculate the correlation coefficient, r, using the following equation:

I(x - X)(y - fj) (1-31)r = -V-;:I=(x=-::::::::X1'~I(=y~-=:Yf:=;;

When there is perfect correlation between the two variables (i.e., a perfect linear relationship), r == 1. When the two variables are completely independent, r = O. Depending on the degrees of freedom and the

16 Physical Pharmacy

~ yintercept = 1.502 1.500 ",

"

= o Slope = -4.325 X10-4 ~ 1.490 .~ tl l!...

o

'IIa::

1.480 o

o Equation of line y = -4.325 X 10-4x + 1.502

1.470 0 20 40 60

Carbon tetrachloride, % by volume Fig. 1- 7. Slope, intercept, and equation of line for data in Table 1-11 calculated by regression analysis.

chosen probability level, it is possible to calculate values of r above which there is significant correlation and below which there is no significant correlation. Obviously, in the latter case, it is not profitable to proceed further with the analysis unless the data can be plotted in some other way that will yield a linear relation. An example of this is shown in Figure 1-4, in whicha linear plot is obtained by plotting the logarithm of oil sepration from an emulsion against emulsifier concentration, as opposed to Figure 1-3, in which the raw data are plotted in the conventional manner. Assuming that the calculated value of r shows a

significant correlation between x and y, it is then necessary to calculate the slope and intercept of the line using the equation:

b = I(x - x)(y - 'Ii) (1-32) I(x - ~)2

in which b is the regression coefficient, or slope. By substituting the value for b in equation (1-33), we can then obtain the y intercept:

'Ii = y + b(x - x) (1-33) The following series of calculations, based on the data in Table 1-11, will illustrate the use of these equations. Exampl, 1-6. Using the data in Table 1-11, calculate the

correlation coefficient, the regression coefficient, and the intercept on the y axis: Examination of equations (1-31), (1-32), and (1-33) shows the

various values we must calculate, and these lire set up as shown:

x (x - x) (x .; X)2 10.0 -26.0 676.0 26.0 -10.0 100.0 33.0 - 3.0 9.0 50.0 +14.0 196.0 61.0 +25.0 625;0

I = 180.0 I=O I = 1606.0 x= 36.0

Y (y - y) (y - yf 1.497 +0.011 0.000121 1.493 +0.007 0.000049 1.485 -0.001 0.000001 1.478 -0.008 0.000064 1.477 -0.009 0.000081

I = 7.430 I~ I = 0.000316 Y= 1.486

(x - X)(y - Y> -0.286 -0.070 +0.003 -0.112 -0.225

I = -0.693

Substituting the relevant values into equation (1-31) gives

r = -0.693 = -0.97* \/1606:0 x 0.000316

From equation (1-32) b = -0.693 = -4 315 x 10-4

1606.0 . and finally, from equation (1-33)

the intercept on the yaxis = 1.486 -4.315 x 10-4 (0 - 36)

= +1.502 Note that for the intercept, we place x equal to zero in equation (1-31). By inserting an actual value of x into equation (1-33), we obtain the value ofy that should be found at that particular value of x. Thus, when x = 10,

Y = 1.486 - 4.315 x 10-4 (10 - 36) = 1.486 - 4.315 x 10-4 (-26) = 1.497

The value agrees with the experimental value, and hence this point lies on the statistically calculated slope drawn in Figure 1-7. Multiple Linear and Polynomial Regression. Regression

for two, three, or more independent variables may be performed, using a linear equation:

y = a + bXl + CX2 + dX3 + . . . (1-34)

"'The theoretic values of r when the probability level is set at 0.05 are:

Degrees of freedom (N - 2): 2 3 5 10 20 50

Correlation coefficient, r: 0.95 0.88 0.75 0.58 0.42 0.27

In Example 1-6, (N - 2) = 3 and hence the theoretic value of r = 0.88. The calculated value was found to be 0.97, and the correlation between x and y is therefore significant.

For a power series in x, a polynomialform is employed: y == a + bx + cx2 + dx3 + . . . (1-35)

In equations (1-34) and (1-35), y is the dependent variable, and a, b, c, and d are regression coefficients obtained by solving the regression equation. Computers are used to handle these more complex equations, but some hand-held calculators, such as the Hewlett Packard HP41C, are programmed to solve multiple regression equations containing two or more independent variables. The r used in multiple regression is called the square of the multiple correlation coefficient and is given the symbol R2 in some texts to distinguish it from r, the square of the linear correlation coefficient. Multiple regression analysis is treated by Draper and Smith. 10

References and Notes 1. S. W. Goldstein, A. M. Mattocks, and U. Beirrnacher, J. Am.

Pharm. Assoe., Pract. Ed. 12,421, 1951. 2. S. W. Goldstein and A. M. Mattocks, Professumai Equilibrium

and Compounding Precision, a bound booklet reprinted from the J. Am. Pharm. Assoc., Pract. Ed. April-August 1951.

3. CRC Handbook of Chemistry and Physics, 63rd Edition, CRC Press, Boca Raton, FL., 1982-83, p. D-227.

4. F. Daniels, Mathe'TfUJ,tical Preparation for Physical Chemistry, McGraw-Hill, New York, 1928.

5. E. A. Brecht, in Sprowls' American PhaTrru;tCy, L. W. Dittert, Ed., 7th Edition, Lippincott, Philadelphia, 1974, Chapter 2.

6. W. J. Youden, Statistical Methods for Chemists, R. Krieger, Huntington, N.Y., Reprint of original edition; 1977, p. 9.

7. S. W. Goldstein, J. Am. Pharm. Assoc., Sci. Ed. 38, 18, 131, 1949; ibid., 39, 505, 1950.

8. L. Saunders and R. Fleming, Mathe'TfUJ,tics and Statistics, Pharmaceutical Press, London, 1957.

9. L. E. Bingenheimer, Am. J. Pharm. Educ. 17,236, 1953. 10. N. Draper and H. Smith, Applied Regression ArULlysis, Wiley,

New York, 1980. 11. N. C. Harris and E. M. Hemmerling, Introductory Applied

Physics, McGraw-Hill, New York, 1972, p. 310. 12. M. J. Kamlet, R. M. Doherty, V. Fiserova-Bergerova, P. W.

Carr, M. H. Abraham and R. W. Taft, J. Pharm. Sci. 76, 14, 1987.

13. W. Lowenthal, "Methodology and Calculations," in Remington's Pharmaceutical Science, A. R. Gennaro, Editor, Mack Publishing, Easton, PA., 1985, p. 82. I

Probl,ms 1-1. The density p of a plastic latex particle is 2.23 g cm",

Convert this value into SI units. Answer: 2.23 x lOS kg m-3 (See Table 1-3 and the physical

constants on the inside front cover of the book for cgs and SI units.) 1-2. Convert 2.736 nm to cm. Answer: 2.736 x 10-7 em 1-3. Convert 1.99 x 104 As into (nancmeters)", Answer: 19.9 nm" 1-4. The surface tension (oy) of a new synthetic oil has a value of

27.32 dyne em". Calculate the corresponding oy value in SI units. Answer:-O.02732 N m- I = 0.02732J m-2 1-5. The work done by the kidneys in transforming 0.1 mole of

urea from the plasma to the urine is 259 cal. Convert this quantity into SI units.

Answer: 1084 J 1-6. The body excretes HCl into the stomach in the concentration

of 0.14 M at 37" C. The work done in this process is 3.8 x 1011 erg. Convert this energy into the fundamental SI units of kg m2 S-2. M stands for molarity.

Answer: 3.8 x 104 kg m2 S-2 1-7. The gas constant R is given in SI units as 8.3143 J OK-I

mole-I. Convert this value into calories. Answer: 1.9872 cal OK-I mole"!

-~...... .

Mereury

76 em Hg.

Pressure of 1 atmosphere

~L-~ ....

.....J '"

)

Fig. 1-8. Owing to atmospheric pressure, mercury rises to a height of 76 em, as demonstrated here.

1-8. Convert 50,237 Pa to torrs or mm Hg, where 1atm = 760mm Hg = 760 torrs.

Answer: 376.8 torrs 1-9. Convert an energy of 4.379 x 106 erg into SI units. Answer: 0.4379 J 1-10. A pressure of 1 atmosphere will support a column of

mercury 760 mm high at 0 C (Fig. 1-8). To what height will a pressure of 1 atm support a column of mineral spirits at 25 C? The density of mineral spirits (mineral oil fraction) at 25 C is 0.860 g/cm", Express the results both in feet and in millimeters of mineral spirits. Hint: See Example 1-1.

Answer: 12010 mm of mineral spirits (or 39 ft) 1-11. How high can an ordinary hand-operated water pump (Fig.

1-9)lift a column of water at 25C above its surface from a water well? How high could it lift a column ofmercury at 25 C from a well filled with mercury? The density of mercury at 25 C is 13.5340 g em".

Answer: According to Harris and Hemmerling,'! "since atmospheric pressure at the sea level is approximately 34 ft of water the most perfect pump of this type could not lift water more than 34 ft from the water level in the well." The same argument applies in the case of mercury.

1-12. Derive equation (1-15), shown on page 6 and used in this problem. Choose a base b for a logarithmic system. For the fun of it, you may pick a base b = 5.9 just because your height is 5.9 feet. Set up a log table with b = 5.9 for the numbers 1000, 100, 10,0.1, and 0.001.

Answer: For the number 0.001 you would obtain log5.9(0.OOI) = -3.8916

1-13. According to Boyle's law of ideal gases, the pressure and volume of a definite mass of gas at a constant temperature are given by the equation PV =k or P =k(llV), in which P is the pressure in atmospheres and V is the volume in liters. Plot the tabulated data so as to obtain a straight line and find the value of the constant k from the graph. Express the constant in ergs, joules, and calories.

Data for Problem 1-13

P (atm) 0.25 0.50 1.0 2.0 4.0 V (liters) 89.6 44.8 22.4 11.12 5.60

Answer: k = 22.4 liter atm, 2.27 x 1010 ergs, 2.27 x 103 joules, 5.43 x 102cal

1-14. The distance traveled by a free-falling body released from rest is given by the equation s = (112)gt2. Plot the accompanying data so as to obtain a straight line and determine the value of g, the

17

18 Physical Pharmacy

If Downstroke

Atmospheric I Pressure t

Water

Fil. 1-9. An old-fashioned hand-operated water pump, showing the piston and valves required to lift the water from the well.

acceleration due to gravity. From the graph, obtain the time that has elapsed when the body has fallen 450 feet.

Data for Problem 1-14

s (ft) 0 16 64 144 256 400 t (sec) 0 1 2 3 4 5

Answer: g = 32 ftJsecj t = 5.3 sec 1-15. The amount of acetic acid adsorbed from solution by charcoal

is expressed by the Freundlich equation, x/m = kc", in which x is the millimoles of acetic acid .adsorbed by m grams of charcoal when the concentration of the acid in the solution at adsorption equilibrium is c molelliter, and k and n are empirical constants. Convert the equation to the logarithmic form, plot log(x/m) vs. log c, and obtain k and n from the graph.

Data for Problem 1-15

Millimoles of acetic Concentration of acid per gram of acetic acid charcoal (x/m) (molelliter) (c)

0.45 0.018 0.60 0.03 0.80 0.06 1.10 0.13 1.50 0.27 2.00 0.50 2.30 0.75 2.65 1.00

Answer: k = 2.65; n = 0.42 1-16. The equation describing the effect of temperature on the

rate of a reaction is the Arrhenius equation,

E.

k = Ae RT 0-36) in which k is the reaction rate constant at temperature T (in degrees absolute), A is a constant known as the Arrhenius factor, R is the gas constant (1.987 cal mole"? deg"), and Ea is known as the energy of activation in cal mole -I. Rearrange the equation so as to fonn an equation for a straight line (see equation (1-19 and then calculate Ea and A from the following data:

Data for Problem 1-16

40 50 60

0.10 0.25 0.70

Hint: Take the natural logarithm of both sides of the Arrhenius equation.

Answer: E. = 20,152 cal mole": A = 1.13 x 1013 sec-I 1-17. (a) Convert the equation, FN =",' G into logarithmic form,

where F is the shear stress or force per unit area, G is rate of shear, and ",' is a viscosity coefficient. N is an exponent that expresses the deviation of some solutions and pastes from the Newtonian viscosity equation (see Chapter ir, in particular pp. 454 and 456). Plot log G versus log F (common logarithms) from the table of experimental data given below, and solve for N, the slope of the line. Onthe same graph plot In G vs. InF (natural logarithms) and again obtain the slope, N. Does the value of N differ when obtained from these two lines?

Chapter 1 Introduction 19

(b) Directly plot the F and G values of the table on l-eycle by 2-cycle log-log paper (i.e., graph paper with a logarithmic scale on both the horizontal and the vertical axes). Does the slope of this line yield the coefficient N obtained from the slope of the previous two plots? How can one obtain N using log-log paper? (c) Regress In G vs. In F and log G vs. log F and obtain N from the

slope of the two regression equations. Which method, (a), (b), or (c), provides more accurate results?

Data for Problem 1-17

G (sec") 22.70 45.40 68.0 106.0 140.0 181.0 272.0

F (dyne/cm'') 1423 1804 2082 2498 2811 3088 3500

Hint: Convert G and F to In and to log and enter these in a table of G and F values. Carry out the log and In values to four decimal places. Partial Arnrwer: (c) log G = 2.69 log F - 7.1018; r = 0.9982

N = 2.69, lJ' = (antilog 7.1018) = 1.26 x 107 In G = 2.69 In F - 16.3556; r = 0.9982 N = 2.69; lJ' = antiln(16.3556) = 1.27 x 107

1-18. In the equation

ds -- = ks (1-37)dt

s is the distance a car travels in time t and k is a proportionality constant. In this problem the velocity dsldt is proportional to the distance s to be covered at any moment. Thus the speed is not constant, but rather is decreasing as the car reaches its destination, probably because the traffic is becoming heavier as the car approaches the city. The equation may be integrated to solve for k, knowing the distance s at time t and the total distance, So. Separating the variables and integrating between the limits of s = s; at t = 0 and s = s at t = t yields

ds ftfs - -=k dt (1-38) s" s 0

(-In s) - (-In s.) = kt and In s = In s" - kt, or in terms of common logs, log s = log s, - 2.;~3' The quantity 2.303 must be used because In s = 2.303 log s, as shown on p. 5. Knowing the remaining distance, 8, at several times, one obtains k

from the slope, and the total distance So from the intercept

Data for Problem 1-18

8 (km) 259.3 192.0 142.3 105.4 t (hr) 1 2 3 4

Compute So and k using In and log values with the data given in the table above. Discuss the advantages and disadvantages in using natural logarithms (In) and common logarithms (log) in a problem of this kind. Why would one c